NORMALIZED SOLUTIONS OF SUPERCRITICAL NONLINEAR FRACTIONAL SCHRODINGER EQUATION WITH POTENTIAL

We are concerned with the following nonlinear fractional Schrodinger equation: (-Delta)(s)u + V(x)u + omega u = vertical bar u vertical bar(p-2)u in R-N, (P) where s is an element of (0, 1) and p is an element of (2 + 4s/N, 2(s)*), that is, the mass supercritical and Sobolev subcritical. Under certain assumptions on the potential V : R-N -> R, positive and vanishing at infinity including potentials with singularities (which is important for physical reasons), we prove that there exists at least one L-2- normalized solution (u, omega) is an element of H-s(R-N) x R+ of equation (P). In order to overcome the lack of compactness, the proof is based on a new min-max argument and splitting lemma for nonlocal version.